Low regularity a priori bounds for the modified Korteweg-de Vries equation

Abstract

We study the local well-posedness in the Sobolev space H^{s}(R) for the modified Korteweg-de Vries (mKdV) equation ∂_{t}u+∂_{x}³u±∂_{x}u³=0 on R. Kenig-Ponce-Vega <cite>KPV2</cite> and Christ-Colliander-Tao <cite>CCT1</cite> established that the data-to-solution map fails to be uniformly continuous on a fixed ball in H^{s}(R) when s<(1/4). In spite of this, we establish that for -(1/8)<s<(1/4), the solution satisfies global in time H^{s}(R) bounds which depend only on the time and on the H^{s}(R) norm of the initial data. This result is weaker than global well-posedness, as we have no control on differences of solutions. Our proof is modeled on recent work by Christ-Colliander-Tao <cite>CCT2</cite> and Koch-Tataru <cite>KT</cite> employing a version of Bourgain's Fourier restriction spaces adapted to time intervals whose length depends on the spatial frequency.